Declared example preparations and thermodynamic interpretation

Here $\log$ denotes the natural logarithm defined in The natural logarithm as the inverse of the exponential function.

For each named finite classical model in this page, adopt the preparation explicitly stated in its theorem. In a canonical branch, take a measurable phase space with a specified reference measure $\nu$, a measurable Hamiltonian $H$, an inverse energy parameter $\beta >0$, and a finite positive partition function $Z=\int e^{-\beta H}d\nu$; the adopted probability measure is $dP=Z^{-1}{e}^{-\beta H}d\nu$. In a grand-canonical branch, take disjoint particle-number sectors with sector partition functions $Z_N$ and a dimensionless activity $z>0$ for which $\mathrm{\Xi }={\sum }_{N\ge 0}{z}^N{Z}_N$ is finite; sector $N$ has probability weight $z^N{Z}_N\mathrm{/}\mathrm{\Xi }$. The spherical-shell and product preparations are separate explicitly specified probability laws in their consuming theorem statements. These are alternative model assumptions, not consequences of a Hamiltonian flow and not assertions that a laboratory system realizes them.

For a canonical model whose reference measure and Hamiltonian are fixed, adopt the statistical thermodynamic definitions $U={\mathbb{E}}_P[H]$, $F=-{\beta }^{-1}\log Z$, and $S=k_B(\log Z+\beta U)$ when the displayed integrals are finite. For a volume-dependent model, define the model pressure as $-{\mathrm{\partial }}_{V}F$ only along the declared fixed-shape volume family. Here $k_B>0$ has SI units J K${}^{-1}$, temperature $T>0$ has units K, and $\beta =(k_{B}T{)}^{-1}$ has units J${}^{-1}$. The fixed reference measure is part of the preparation; changing it changes the numerical partition function and entropy convention.

For the continuous particle examples, adopt the operational species and counting convention of Classical particle counting and physical observables. Use one fixed primitive action reference $a_0>0$ in J s, and write $h_0:=a_0$ where the consuming model uses the notation $h_0$. In a Cartesian physical phase chart $dqdp$ has action dimension $(\mathrm{J}\mathrm{s}{)}^{dN}$, so the reference $dqdp\mathrm{/}({\prod }_s{N}_s!a_0^{dN})$ is dimensionless. Each model retains its declared operational labels: an omitted factorial for resolved harmonic coordinates is not replaced by a particle-exchange quotient. The finite spin examples use their stated discrete counting measure, without a continuous phase-cell factor. Choosing a numerical reference action, or identifying it with Planck's constant, is an additional convention; no such value or identification is imposed here. The Gaussian and gamma identities in Gaussian, gamma and beta product integrals establish the mathematical normalizations, not the physical preparation or empirical equilibration.

A finite partition function and normalized density are mathematical conditions of each canonical consumer. No negative-temperature unbounded gas, ergodic hypothesis, observed outcome, or universal ensemble-selection principle is included.